Hún bætir einni tilvitnun við innganginn og annarri í miðjuna.

Bætir is the 3rd person singular present tense of að bæta.

So Hún bætir ... means She adds ... or She is adding ..., depending on context.

• hún = she
• bætir = adds

The ending -ir is a common present-tense ending for many Icelandic verbs in the he/she/it form.

Because að bæta normally takes the thing being added in the dative case.

So in this sentence:

• einni = dative singular feminine of einn
• tilvitnun is also singular feminine, and in this noun its indefinite singular form looks the same in several cases

That is why you get:

• einni tilvitnun = one quotation in the dative

A learner might expect accusative, but with að bæta einhverju ... Icelandic uses the dative for the thing that is added.

They are both dative singular feminine forms, agreeing with tilvitnun, which is a feminine singular noun.

• ein(n) → einni in dative feminine singular
• annar → annarri in dative feminine singular

So both words match the grammatical gender, number, and case of the noun they go with.

This is one of the big things English speakers have to get used to in Icelandic: words like one and another change form to agree with the noun.

Because this noun happens to have the same indefinite singular form in more than one case.

So even though tilvitnun is dative here, it still appears as tilvitnun.

That means the case is shown more clearly by the matching word before it:

• einni tilvitnun

In other words, the dative is visible in einni, even though tilvitnun itself does not look different.

Annarri means another or a second one here.

The noun tilvitnun is simply omitted because it is understood from the first half of the sentence. This is very natural in Icelandic.

So the full expanded meaning is:

• Hún bætir einni tilvitnun við innganginn og annarri tilvitnun í miðjuna.

But Icelandic often leaves out the repeated noun when it is obvious.

Because the two phrases are expressing slightly different kinds of destination.

• við innganginn = to the introduction
• í miðjuna = into the middle

With að bæta, við is very common for the idea add X to Y.

But í is natural when the idea is putting something into a place or section, especially with something like miðjan = the middle.

So the sentence mixes two natural Icelandic ways of expressing where the quotation is being added:

• one is added to the introduction
• another is added into the middle

Because the sentence refers to specific parts of something:

• innganginn = the introduction
• miðjuna = the middle

In English, we also usually say the introduction and the middle in this kind of context, not just an introduction or a middle.

Their forms are:

• inngangur → innganginn
• miðja → miðjuna

These are the definite singular forms.

Because í can take either accusative or dative, depending on meaning.

• accusative = movement/direction into something
• dative = location in something

Here the meaning is directional: she is adding a quotation into the middle. So Icelandic uses:

• í miðjuna = into the middle

If it meant location, as in it is in the middle, you would expect:

• í miðjunni = in the middle

This is a very common Icelandic pattern with prepositions.

Here it is most naturally understood as one, because it is paired with annarri = another.

So the structure is:

• one quotation ... and another ...

If the sentence only wanted to say a quotation, Icelandic could sometimes omit einn/ein/eitt entirely, depending on style and context.

But here the contrast between one and another is important, so einni ... og annarri ... is a very natural choice.

Because inngangur can mean both entrance and introduction, and the context tells you which one is meant.

Here the sentence talks about adding quotations and also mentions the middle, so the topic is clearly a text or document. In that context:

• inngangurinn = the introduction

If the context were about a building, then inngangur would more likely mean entrance.